| L(s) = 1 | + 1.20·2-s − 0.557·4-s + 3.21·5-s + 7-s − 3.07·8-s + 3.86·10-s − 4.57·11-s + 0.703·13-s + 1.20·14-s − 2.57·16-s − 4.25·17-s + 8.04·19-s − 1.79·20-s − 5.49·22-s + 5.34·23-s + 5.34·25-s + 0.844·26-s − 0.557·28-s + 5.39·29-s + 7.61·31-s + 3.05·32-s − 5.10·34-s + 3.21·35-s + 5.19·37-s + 9.66·38-s − 9.88·40-s + 41-s + ⋯ |
| L(s) = 1 | + 0.849·2-s − 0.278·4-s + 1.43·5-s + 0.377·7-s − 1.08·8-s + 1.22·10-s − 1.38·11-s + 0.194·13-s + 0.320·14-s − 0.643·16-s − 1.03·17-s + 1.84·19-s − 0.401·20-s − 1.17·22-s + 1.11·23-s + 1.06·25-s + 0.165·26-s − 0.105·28-s + 1.00·29-s + 1.36·31-s + 0.539·32-s − 0.876·34-s + 0.543·35-s + 0.853·37-s + 1.56·38-s − 1.56·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.131592993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.131592993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 5 | \( 1 - 3.21T + 5T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 - 0.703T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 - 8.04T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 - 5.39T + 29T^{2} \) |
| 31 | \( 1 - 7.61T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 + 3.09T + 67T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 - 4.13T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.938173222867305431161031893319, −8.237356294700823802115248995340, −7.19341402616786774086767751187, −6.27508697325133296710043117626, −5.54878683441600860659104232077, −5.07929170631309858962992039037, −4.37963790506389762853610374524, −2.93634461950996874307253677363, −2.54145704620554990643730587048, −1.02876919361776193338675305527,
1.02876919361776193338675305527, 2.54145704620554990643730587048, 2.93634461950996874307253677363, 4.37963790506389762853610374524, 5.07929170631309858962992039037, 5.54878683441600860659104232077, 6.27508697325133296710043117626, 7.19341402616786774086767751187, 8.237356294700823802115248995340, 8.938173222867305431161031893319
